Monday, September 17, 2012

A vector field is a function that assigns a vector value to each point in 3D space. Vector fields can be used to represent things like wind (the vector field specifies the wind velocity at each point in space), water, magnetism, etc.

To me, wind is the most interesting use case. I want a system that can be used for physics (trees, tumble weed, paper cups), particles (leaves, sparks, smoke) and graphics (grass). I also want the system to be capable of handling both global effects (wind blowing through the entire level) and local effects (explosions, air vents, landing helicopters, rising hot air from fires, etc). But I don't want to limit the system to only handling wind. I imagine that once the system is in place, it could be put to other interesting uses as well.

There are a number of things that make this an interesting non-trivial design challenge:

Vector fields represent a global shared state. All systems (particles, physics, etc) should react to the same wind. This can create strong couplings between unrelated systems, which we want to avoid.

The system must be fast. We want to be able to make large particle effects that are affected by wind. As a design goal, let's say that it should be able to handle at least 10 000 queries / frame.

As stated above, the system must be flexible enough to handle both global wind and a large variety of different local effects (air vents, fans, etc).

I'll outline the system in a series of articles. Let's start by thinking a bit about how we can represent the vector field in a way that allows for fast queries.

1. Use a functional representation

Storing the vector value for every point in 3D space at a decent resolution would require huge amounts of memory. It would also be very expensive to update. If we wanted to change the global wind direction, we would have to loop over all those points and change the value.

So, instead, we will use a functional representation. We will express the field as some closed function F(p, t) that gives us the field vector at point p in space at the time t.

For example, we could express a global wind that oscillates in the x-direction as:

F(p, t) = Vector3(sin(t), 0, 0)

The closed function form allows us to evaluate the vector field at any point in space and time.

Note that even with a functional form as the main representation, we can still interact with grid based representations. For example, we can render some section of the F(p, t) function to a texture for use on a GPU. Similarly, if we have some grid based wind data that we want to add to the simulation, we could use that as part of the F(p, t) expression:

F(p, t) = Vector3(sin(t), 0, 0) + sample_grid(grid, p)

2. Ignore the time coordinate

The vector field function F(p, t) is a function of both space and time. The wind varies throughout the level and if we look at any one point, the wind at that point varies over time.

But in practice, we treat the p and t coordinates very differently. We start at some time t_0 and then evaluate F(p, t_0) for thousands of different p values. Then we move on to t_1 and do the same thing.

We can make use of the fact that t remains constant for a large number of evaluations to simplify the function. For example at t=0.5 the function:

F(p, t) = sin(p.x) * sin(p.y) * cos(t)

simplifies to:

G(p) = sin(p.x) * sin(p.y) * 0.8776

which is cheaper to evaluate.

Taking this approach a step further, it makes sense to split our system in two parts -- a high level system that knows about time and every frame produces a new G(p) for the current time, and a low level system that ignores time completely and just computes G(p). Since the high level system only runs once per frame it can afford to do all kinds of complicated but interesting stuff, like constant folding, optimization, etc.

For the low level system we have reduced the problem to evaluating G(p).

3. Express the field as a superposition of individual effects

To make it possible for the field to contain both global effects (world wind) and local effects (air vents, explosions) we express it as a superposition of individual effect functions:

G(p) = G_1(p) + G_2(p) + ... + G_n(p)

Here G_i(p) represents each individual effect. A base wind could be expressed as just a constant:

G_0(p) = Vector3(2.1, 1.4, 0)

A turbulence function could add a random component

G_1(p) = turbulence(seed, p, 4)

An explosion effect could create a wind with a speed of 100 m/s outwards from the center of the explosion in a sphere with radius 4.0 meter around the explosion center:

G_2(p) = sphere(p,c,4) * normalize(p-c) * 100

Here sphere(p,c,4) is a spherical support function that defines the range of the effect. It is 1 if ||p - c|| <= 4.0 and 0 otherwise.

Note again that we have stripped out the time component. At the higher level, this might be an expanding sphere with decreasing wind speeds, but at the low level we only care what it looks like at this instance.

Similar functions can be added for other local effects.

4. Use the AABB to cull local fields

If we have a lot of local effects (explosions, etc), evaluating G(p) will be pretty expensive.

We can reduce the cost by only evaluating the local effects that are close enough to our particle system to matter.

I.e., instead of evaluating G(p) for all particles, we first intersect the AABB of each G_i(p)'s support with the AABB of our particle system.

That gives us a simpler function G'(p) that we can then evaluate for each particle.

If we wanted to, we could use the wavelength of the field for further simplifications. If the scale at which a field effect changes is much larger than our AABB, we can replace that effect with a Taylor series expansion. Similarly, if an effect oscillates at a scale much smaller than the size of our particles, we can replace it with its average value.

Next time

Next time I will look at how we can efficiently evaluate arbitrary functions, such as:

Monday, September 3, 2012

Recently, I've become increasingly dissatisfied with the standard C++ way of organizing header files (one .h file and one .cpp file per class) and started experimenting with alternatives.

I have two main problems with the ways headers are usually organized.

First, it leads to long compile times, especially when templates and inline functions are used. Fundamental headers like array.h and vector3.h get included by a lot of other header files that need to use the types they define. These, in turn, get included by other files that need their types. Eventually you end up with a messy nest of header files that get included in a lot more translation units than necessary.

Sorting out such a mess once it has taken root can be surprisingly difficult. You remove an #include statement somewhere and are greeted by 50 compile errors. You have to fix these one by one by inserting missing #include statements and forward declarations. Then you notice that the Android release build is broken and needs additional fixes. This introduces a circular header dependency that needs to be resolved. Then it is on to the next #include line -- remove it, rinse and repeat. After a day of this mind-numbingly boring activity you might have reduced your compile time by four seconds. Hooray!

Compile times have an immediate and important effect on programmer productivity and through general bit rot they tend to grow over time. There are many things that can increase compile times, but relatively few forces that work in the opposite direction.

It would be a lot better if we could change the way we work with headers, so that we didn't get into this mess to begin with.

My second problem is more philosophical. The basic idea behind object-oriented design is that data and the functions that operate on it should be grouped together (in the same class, in the same file). This idea has some merits -- it makes it easier to verify that class constraints are not broken -- but it also leads to problems. Classes get coupled tightly with concepts that are not directly related to them -- for example things like serialization, endian-swapping, network synchronization and script access. This pollutes the class interface and makes reuse and refactoring harder.

Class interfaces also tend to grow indefinitely, because there is always "more useful stuff" that can be added. For example, a string class (one of my pet peeves) could be extended with functionality for tokenization, path manipulation, number parsing, etc. To prevent "class bloat", you could write this code as external functions instead, but this leads to a slightly strange situation where a class has some "canonized" members and some second-class citizens. It also means that the class must export enough information to allow any kind of external function to be written, which kind of breaks the whole encapsulation idea.

In my opinion, it is much cleaner to organize things by functionality than by type. Put the serialization code in one place, the path manipulation code in another place, etc.

My latest idea about organization is to put all type declarations for all structs and classes in a single file (say types.h):

Note that types.h has no function declarations, but it includes the full data specification of any struct or class that we want to use "by value". It also has forward declarations for classes that we want to use "by reference". (These classes are assumed to have pure virtual interfaces. They can only be created by factory functions.)

Since types.h only contains type definitions and not a ton of inline code, it ends up small and fast to compile, even if we put all our types there.

Since it contains all type definitions, it is usually the only file that needs to be included by external headers. This means we avoid the hairy problem with a big nest of headers that include other headers. We also don’t have to bother with inserting forward declarations in every header file, since the types we need are already forward declared for us in types.h.

We put the function declarations (along with any inline code) in the usual header files. So vector3.h would have things like:

.cpp files that wanted to use these operations would include vector3.h. But .h files and other .cpp files would not need to include the file. The file gets included where it is needed and not anywhere else.

Note that types.h only contains the constructor and the destructor for Array<T>, not any other member functions.

Furthermore, I prefer to design classes so that the "zero-state" where all members are zeroed is always a valid empty state for the class. That way, the constructor becomes trivial, it just needs to zero all member variables. We can also construct arrays of objects with a simple memset().

If a class needs a more complicated empty state, then perhaps it should be an abstract interface-class instead of a value class.

Since the “open structs” in types.h can be accessed from anywhere, we can grop operations by what they do rather than by what types they operate on. For example, we can put all the serialization code in serialization.h and serialization.cpp. We can create a file path.h that provides path manipulation functions for strings.

An external project can also "extend" any of our classes by just writing new methods for it. These methods will have the same access to the Vector3 data and be called in exactly the same way as our built-in ones.

The main drawback of this model is that internal state is not as "protected" as in standard object-oriented design. External code can "break" our objects by manipulating members directly instead of using methods. For example, a stupid programmer might try to change the size of an array by manipulating the _size field directly, instead of using the resize() method.

Naming conventions can be used to mitigate this problem. In the example above, if a type is declared with class and the members are preceded by an underscore, the user should not manipulate them directly. If the type is declared as a struct, and the members do not start with an underscore, it is OK to manipulate them directly. Of course, a stupid programmer can still ignore this and go ahead and manipulate the members directly anyway. On the other hand, there is no end to the things a stupid programmer can do to destroy code. The best way to protect against stupid programmers is to not hire them.

I haven’t yet written anything really big in this style, but I've started to nudge some files in the Bitsquid codebase in this direction, and so far the experience has been positive.